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自转当玩具陀螺的自转轴倾斜时，因其重心不在支点上，所以重力便对支点形成重力矩 MG=r×GM_G=r×GMG​=r×G
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自转当玩具陀螺的自转轴倾斜时，因其重心不在支点上，所以重力便对支点形成重力矩 MG=r×GM_G=r×GMG​=r×G
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                                <h2>
                                    3.2 陀螺仪进动现象
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                                    2023-10-20, 466 words, 2 min read
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                                        <h2 id="一-陀螺的进动现象">一、陀螺的进动现象</h2>
<h4 id="1-受力分析">1、受力分析</h4>
<p>玩具陀螺绕其自转轴进行高速旋转，自转角速度为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></p>
<p>自转当玩具陀螺的自转轴倾斜时，因其重心不在支点上，所以重力便对支点形成重力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>G</mi></msub><mo>=</mo><mi>r</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">M_G=r×G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">G</span></span></span></span></p>
<h4 id="2-力矩效果">2、力矩效果</h4>
<p>重力矩<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>G</mi></msub></mrow><annotation encoding="application/x-tex">M_G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">G</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>始终垂直于矢量 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span> 和重力 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">G</span></span></span></span> 所构成的平面(垂直向外),又因为玩具陀螺的角动量 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span></span></span></span> 与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span> 同向.</p>
<p>所以<strong>重力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">M_c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 始终垂直于玩具陀螺的角动量 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span></span></span></span></strong>。</p>
<h2 id="二-陀螺仪测量角速度原理">二、陀螺仪测量角速度原理</h2>
<figure data-type="image" tabindex="1"><img src="http://cos.pansis.site/202310201933056.png/abc123" alt="image-20231020193354766" loading="lazy"></figure>
<h2 id="三-陀螺仪的表观进动">三、陀螺仪的表观进动</h2>
<h4 id="1-原因">1、原因</h4>
<p>陀螺仪的定轴或方向稳定是相对惯性空间的。</p>
<p>而地球以其自转角速度绕极轴相对惯性空间转动。</p>
<p>因而陀螺自转轴相对地球的方向将出现表观变化，</p>
<h4 id="2-定义">2、定义</h4>
<p>观察者以地球作为参考基准所看到的陀螺仪的自转轴相对地球的方向出现的进动现象，叫做陀螺仪的表观进动。</p>
<h4 id="3-周期-进动轴">3、周期、进动轴</h4>
<p>24小时</p>
<p>逆时针</p>
<img src="http://cos.pansis.site/202310201935229.png/abc123" alt="image-20231020193500171" style="zoom:33%;" />
<img src="http://cos.pansis.site/202310201935154.png/abc123" alt="image-20231020193510075" style="zoom:33%;" />
<h2 id="四-单自由度陀螺仪的特性">四、单自由度陀螺仪的特性</h2>
<h4 id="1-特性">1、特性</h4>
<p>1、当基座绕自转轴 (Y)  或框架轴 (X) 方向转动时，都不会直接带动转子。</p>
<p>2、当基座绕缺少自由度的方向 (Z)转动时，将带动框架和转子一起在Z轴转动。</p>
<p>3、同时，由于Z轴的力矩和陀螺自身动量，产生进动，使陀螺仪绕框架轴 (X) 方向转动，产生的角速率叫做进动角速率。</p>
<img src="http://cos.pansis.site/202310201935941.png/abc123" alt="image-20231020193542889" style="zoom: 50%;" />
<h4 id="2-用途">2、用途</h4>
<p>当基座绕缺少自由度的方向 (Z)转动时，就可以根据单自由度陀螺仪的强迫<strong>进动角速率</strong>来计算得到<strong>基座的转动角速角速率</strong>。</p>
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